Functional calculus for $C_{0}$-groups on real interpolation spaces

$: Functional calculus for $C_{0}$-groups on real interpolation spaces$
FUNCTIONAL CALCULUS ON REAL INTERPOLATION SPACES

FOR GENERATORS OF C0-GROUPS

MARKUS HAASE AND JAN ROZENDAAL

Abstract. We study functional calculus properties of C0-groups on real in-

terpolation spaces, using transference principles. We obtain interpolation ver-

sions of the classical transference principle for bounded groups and of a recenttransference principle for unbounded groups. Then we show that each group

generator on a Banach space has a bounded H∞1 -calculus on real interpolation

spaces. Additional results are derived from this.

1. Introduction

The classical transference principle by Berkson, Gillespie and Muhly from [5]yields an estimate ∥∥∥∥∫

RU(s)xµ(ds)

∥∥∥∥X

≤M2 ‖Lµ‖L(Lp(X)) ‖x‖X(1.1)

for all x ∈ X, where (U(s))s∈R ⊆ L(X) is a bounded C0-group of operators on aBanach space X with uniform bound M , µ is a complex Borel measure on R andLµ is convolution with µ on Lp(X), the space of p-integrable X-valued functions,for p ∈ [1,∞]. Under certain geometrical assumptions on X, the norm of Lµ can bebounded in terms of a suitable norm of the Fourier transform Fµ of µ. For instance,if X is a Hilbert space then ‖Lµ‖L(L2(X)) is equal to ‖Fµ‖∞, by Plancherel’s theo-

rem. If X is a UMD space and p ∈ (1,∞) then bounds for ‖Lµ‖L(Lp(X)) follow from

the Mikhlin multiplier theorem. By combining this with (1.1), functional calculusbounds for the generator A of (U(s))s∈R can be obtained, i.e., estimates of the form‖f(A)‖ ≤ C ‖f‖F for all f in some function algebra F . Such bounds are importantfor evolution equations, conform for instance [2, 17].

Useful as this procedure is, the assumptions on the space X restrict the generalityof the results. In particular, Hilbert and UMD spaces are reflexive. Therefore theapproach described above generally does not yield interesting results for groups ofoperators on non-reflexive spaces, such as C(K)-spaces or L1-spaces. In this paperwe take a different approach and consider transference principles on interpolationspaces. It is known that the functional calculus properties of operators improveupon restriction to interpolation spaces, conform for instance the result of Dore [6]that an invertible sectorial operator has a bounded sectorial H∞-calculus on realinterpolation spaces. However, we are interested in functional calculus on strips,

Date: October 7, 2014.

2010 Mathematics Subject Classification. 47A60, 47D03, 42B35, 42A45, 46B70.Key words and phrases. Functional calculus, Transference, Operator semigroup, Fourier mul-

tiplier, Interpolation space.The second-named author is supported by NWO-grant 613.000.908 “Applications of Transfer-

ence Principles”.

1

2 MARKUS HAASE AND JAN ROZENDAAL

the more natural choice for group generators. We use that on Besov spaces Fouriermultiplier results hold that do not depend on the geometry of the underlying space[9, 16]. Since Besov spaces are obtained from real interpolation between Lp andSobolev spaces, this fits into the setting of a transference principle on interpolationspaces. In Proposition 3.2 we derive the following version of (1.1) on the realinterpolation space (X,D(A))θ,q from (1.4). For the X-valued Besov space Bθp,q(X)see Section 2.2.

Proposition 1.1. Let X be a Banach space and let θ ∈ (0, 1), p ∈ [1,∞) andq ∈ [1,∞]. Then there exists a constant C ≥ 0 such that the following holds. If −iAgenerate a C0-group (U(s))s∈R on a Banach space X with M := sups∈R ‖U(s)‖ <∞, then ∥∥∥∥∫

RU(s)xµ(ds)

∥∥∥∥(X,D(A))θ,q

≤ CM2 ‖Lµ‖L(Bθp,q(X)) ‖x‖(X,D(A))θ,q

for all complex Borel meaures µ on R and x ∈ (X,D(A))θ,q.

Combining Proposition 1.1 with the aforementioned Fourier multiplier results onBesov spaces yields the following, a consequence of Corollary 3.5.

Corollary 1.2. Let −iA generate a uniformly bounded C0-group (U(s))s∈R on aBanach space X, and let θ ∈ (0, 1), q ∈ [1,∞]. Then there exists a constant C ≥ 0such that∥∥∥∥∫

RU(s)xµ(ds)

∥∥∥∥(X,D(A))θ,q

≤ C sups∈R|Fµ(s)|+ (1 + |s|) |(Fµ)′(s)| ‖x‖(X,D(A))θ,q

for all x ∈ (X,D(A))θ,q and for each µ ∈ M(R) such that Fµ ∈ C1(R) withsups∈R(1 + |s|) |(Fµ)′(s)| <∞.

We also obtain an interpolation version of the transference principle for un-bounded groups from [12], as Proposition 3.1. In terms of functional calculus forthe part of A in (X,D(A))θ,q, these transference principles yield a result for func-tions in the analytic Mikhlin algebra

H∞1 (Stω) :=

f ∈ H∞(Stω)

∣∣∣∣ supz∈Stω

(1 + |z|) |f ′(z)| <∞,(1.2)

endowed with the norm

‖f‖H∞1 (Stω) := sup

z∈Stω

|f(z)|+ (1 + |z|) |f ′(z)| (f ∈ H∞1 (Stω)).(1.3)

Here Stω := z ∈ C | |Im(z)| < ω for ω > 0. Note that definition (1.3) of the normin the analytic Mikhlin algebra is different from that in [12], where the quantity‖f‖ = supz∈Stω |f(z)|+|zf ′(z)| is considered. However, the two norms are equivalenton domains containing zero, and (1.3) is more natural in the setting of transferenceprinciples on (inhomogeneous) Besov spaces, since Fourier multiplier results on suchspaces require an inhomogeneous condition at zero. See also Remarks 3.7 and 4.2.

Our main functional calculus result is as follows. For the group type θ(U) see(2.1), and for a proof of this result see Theorem 4.1.

Theorem 1.3. Let −iA be the generator of a C0-group (U(s))s∈R on a Banachspace X, and let θ ∈ (0, 1), q ∈ [1,∞]. Then the part of A in (X,D(A))θ,q has abounded H∞1 (Stω)-calculus for all ω > θ(U). If (U(s))s∈R is uniformly bounded thenthe constant bounding the H∞1 (Stω)-calculus does not depend on ω > 0.

FUNCTIONAL CALCULUS ON INTERPOLATION SPACES 3

In [12], Theorem 1.3 is obtained for group generators on UMD spaces and func-tional calculus for the operator A itself. Our result shows that on interpolationspaces no assumptions on the geometry of the underlying space are required. Thismeans that even on spaces which are not UMD, such as C(K)-spaces and L1-spaces,one can obtain functional calculus results if one is willing to restrict to interpolationspaces. Moreover, our results reaffirm the philosophy that the functional calculusproperties of an operator improve when restricted to interpolation spaces, as wasalready evidenced for functions on sectors by the result of Dore [6, Theorem 3.2].

From Theorem 1.3 we deduce other functional calculus statements, for sectorialoperators and generators of cosine functions.

Section 2 provides the necessary background on functional calculus and the the-ory of Fourier multipliers on Besov spaces. In Section 3 we establish transferenceprinciples on interpolation spaces, and in Section 4 we prove Theorem 1.3. Section5 contains additional results that can be derived from this.

1.1. Notation and terminology. The natural numbers are N := 1, 2, . . ., andwe write N0 := N ∪ 0. The letters X and Y denote Banach spaces over thecomplex number field, and L(X) is the Banach algebra of all bounded operators onX. The domain D(A) ⊆ X of a closed unbounded operator A on X is a Banachspace when endowed with the norm

‖x‖D(A) := ‖x‖+ ‖Ax‖ (x ∈ D(A)).

The spectrum of A is σ(A) and the resolvent set ρ(A) := C \ σ(A). For z ∈ ρ(A)the operator R(z,A) := (zI−A)−1 ∈ L(X) is the resolvent of A at z.

For p ∈ [1,∞], Lp(R;X) is the Bochner space of equivalence classes of X-valuedLebesgue p-integrable functions on R. The Holder conjugate of p ∈ [1,∞] is p′ andis defined by 1

p + 1p′ = 1. The norm on Lp(R;X) is usually denoted by ‖·‖p. In the

case X = C we will simply write Lp(R) = Lp(R;C).By M(R) we denote the space of complex-valued Borel measures on R with the

total variation norm. For ω ≥ 0 we let Mω(R) consist of those µ ∈ M(R) of theform µ(ds) = e−ω|s|ν(ds) for some ν ∈ M(R), with

‖µ‖Mω(R) :=∥∥∥eω|·|µ

∥∥∥M(R)

.

Note that Mω(R) is a Banach algebra under convolution. A function g such that[s 7→ g(s) eω|s|] ∈ L1(R) is usually identified with its associated measure µ ∈ Mω(R)given by µ(ds) = g(s)ds.

For Ω 6= ∅ open in C we let H∞(Ω) be the unital Banach algebra of boundedholomorphic functions on Ω with the supremum norm

‖f‖H∞(Ω) := supz∈Ω|f(z)| (f ∈ H∞(Ω)).

We mainly consider the case where Ω is a strip of the form

Stω := z ∈ C | |Im(z)| < ω

for ω > 0, with St0 := R.The Schwartz class S(R;X) is the space of X-valued rapidly decreasing smooth

functions on R, and the space of X-valued tempered distributions is S ′(R;X). TheFourier transform of an X-valued tempered distribution Φ ∈ S ′(R;X) is denoted


by FΦ. For instance, if µ ∈ Mω(R) for ω > 0 then Fµ ∈ H∞(Stω)∩C(Stω) is givenby

Fµ(z) :=

∫R

e−iszµ(ds) (z ∈ Stω).

If X and Y are Banach spaces that are embedded continuously into a Hausdorfftopological vector space Z, then we call (X,Y ) an interpolation couple. We let

K(t, z) := inf ‖x‖X + t ‖y‖Y | x ∈ X, y ∈ Y, x+ y = zfor t > 0 and z ∈ X + Y ⊆ Z. The real interpolation space of X and Y withparameters θ ∈ [0, 1] and q ∈ [1,∞] is

(X,Y )θ,q :=z ∈ X + Y | [t 7→ t−θK(t, z)] ∈ Lq((0,∞),dt/t)

,(1.4)

a Banach space when equipped with the norm

‖z‖(X,Y )θ,q:=∥∥t 7→ t−θK(t, z)

∥∥Lq((0,∞),dt/t)

(z ∈ (X,Y )θ,q).

If T : X+Y → X+Y restricts to a bounded operator on X and a bounded operatoron Y then

‖T‖L((X,Y )θ,q)≤ ‖T‖1−θL(X) ‖T‖

θL(Y )(1.5)

for all θ ∈ (0, 1) and q ∈ [1,∞] [4, Theorem 3.1.2]. We mainly consider interpolationspaces for the couple (X,D(A)), where A is a closed operator on X. We write

DA(θ, q) := (X,D(A))θ,q

and

‖x‖θ,q := ‖x‖DA(θ,q) (x ∈ DA(θ, q)).

For an operator B on X and a continuously embedded space Y → X, the operatorBY on Y that satisfies BY y = By for elements in its domain

D(BY ) := y ∈ D(B) ∩ Y | By ∈ Y is the part of B in Y . If Y = DA(θ, q) for θ ∈ (0, 1) and q ∈ [1,∞] then we write

Bθ,q := BDA(θ,q).

Throughout, an X-valued function space Φ(R;X) on the real line will be denotedby Φ(X) whenever little confusion can arise.

2. Functional calculus and Fourier multipliers

2.1. Functional calculus. We assume that the reader is familiar with the basicsof the theory of C0-groups as developed in, for instance, [8], and merely recall someof the notions and results in functional calculus theory that are used. Details onfunctional calculus for group generators can be found in [10, Chapter 4].

Let −iA be the generator of a C0-group (U(s))s∈R on a Banach space X. Thenthe group type of U ,

θ(U) := infω ≥ 0

∣∣∣∃M ≥ 1 such that ‖U(s)‖ ≤Meω|s| for all s ≥ 0,(2.1)

is finite. Moreover, A is a strip-type operator of height ω0 := θ(U), i.e., σ(A) ⊆ Stω0

and

supλ∈C\Stω

‖R(λ,A)‖ <∞ for all ω > ω0.


The strip-type functional calculus for A is defined as follows. First, operators f(A) ∈L(X) are associated with functions

f ∈ E(Stω) :=g ∈ H∞(Stω)

∣∣g(z) ∈ O(|z|−α) for some α > 1 as |Re(z)| → ∞

for ω > ω0, by a Cauchy-type integral

f(A) :=1

2πi

∫δStω′

f(z)R(z,A) dz.

Here δStω′ is the positively oriented boundary of Stω′ for ω′ ∈ (ω0, ω). This proce-dure is independent of the choice of ω′ by Cauchy’s theorem, and yields an algebrahomomorphism E(Stω) → L(X). The definition of f(A) is extended to a largerclass of functions by regularization, i.e.

f(A) := e(A)−1(ef)(A)

if there exists e ∈ E(Stω) with e(A) injective and ef ∈ E(Stω). This yields a closedunbounded operator f(A) on X, and the definition of f(A) is independent of thechoice of the regularizer e. The algebra of all meromorphic functions on Stω thatare regularizable for A is denoted byMA(Stω). Each f ∈ H∞(Stω) is regularizableby the function z 7→ (λ− z)−2, for |Im(λ)| > ω.

Since −iA generates a C0-group, the Hille-Phillips functional calculus for Ayields certain functions f that give rise to bounded operators f(A). Fix M ≥ 1and ω ≥ 0 such that ‖U(s)‖ ≤Meω|s| for all s ∈ R. For µ ∈ Mω(R) define

Uµx :=

∫RU(s)xµ(ds) (x ∈ X).(2.2)

Then µ 7→ Uµ is an algebra homomorphism Mω(R)→ L(X). The following lemma,Lemma 2.2 in [11], shows that the Hille-Phillips calculus extends the strip-typecalculus for A.

Lemma 2.1. Let X, A and U be as above, and let ω′ > ω ≥ 0.

a) For each f ∈ E(Stω′) there exists µ ∈ Mω(R) such that f = Fµ.b) Let µ ∈ Mω(R) be such that Fµ extends to an element of MA(Stω′). Then

f(A) = Uµ ∈ L(X) and

supt∈R‖f(t+A)‖ ≤M ‖µ‖Mω(R) .

We now consider functional calculus for operators on interpolation spaces. Thefollowing lemma shows that, in particular, the functional calculi for A and Aθ,q arecompatible.

Lemma 2.2. Let A be a strip-type operator of height ω0 on a Banach space X andlet θ ∈ (0, 1), q ∈ [1,∞] and m,n ∈ N0. Let Y := (D(Am),D(An))θ,q.

a) The part AY of A in Y is a strip-type operator of height ω0. Moreover,f ∈MAY (Stω) with f(AY ) = f(A)Y for all ω > ω0 and f ∈MA(Stω).

b) If −iA generates a C0-group (U(s))s∈R on X and q <∞, then −iAY gen-erates the C0-group (U(s)Y )s∈R. In particular, D(AY ) is dense in Y .

Proof. a) First note that, for all k ∈ N0 and λ ∈ ρ(A), R(λ,A) leaves D(Ak)invariant with ‖R(λ,A)‖L(D(Ak)) ≤ ‖R(λ,A)‖L(X). By (1.5), R(λ,A) leaves Y

invariant with

‖R(λ,A)‖L(Y ) ≤ ‖R(λ,A)‖L(X) .(2.3)


By [10, Proposition A.2.8], σ(AY ) ⊆ σ(A) and R(λ,AY ) = R(λ,A)Y for all λ ∈ρ(A). Hence (2.3) yields the first statement. Let ω > ω0 and f ∈ E(Stω) be given.Then

f(AY )y =1

2πi

∫Γ

f(z)R(z,AY )y dz =1

2πi

∫Γ

f(z)R(z,A)y dz = f(A)y

for some contour Γ and all y ∈ Y . For a general f ∈ MA(Stω), note that e is aregulariser for f in the functional calculus for AY if it is a regulariser for f in thefunctional calculus for A, since then e(AY ) = e(A)Y is injective. The rest followsby regularization.

b) By (1.5), ‖U(s)Y ‖ ≤ ‖U(s)‖ for all s ∈ R. Hence (U(s)Y )s∈R is locallybounded. Since it is strongly continuous on the dense subset D(Amax(n,m)) ⊆ Y[19, Proposition 1.2.5], it is strongly continuous on Y . By [8, p. 60], −iAY is itsgenerator.

Remark 2.3. Part b) of Lemma 2.2 ensures that the integral in (2.2) is well-definedand converges in DA(θ, q), for x ∈ DA(θ, q) and q < ∞. Even though (U(s))s∈Ris not strongly continuous on DA(θ,∞) in general, the integral is well-defined andconverges in X. Since DA(θ, q) is continuously embedded in X for all θ ∈ (0, 1)and q ∈ [1,∞], the value of the integral does not depend on the space in whichconvergence takes place. Hence from now on we regularly will not specify in whichnorm (2.2) converges.

Let A be a strip-type operator of height ω0 and ω > ω0. For a Banach algebraF of functions that is continuously embedded in H∞(Stω), we say that A has abounded F -calculus if there exists a constant C ≥ 0 such that f(A) ∈ L(X) with

‖f(A)‖L(X) ≤ C ‖f‖F for all f ∈ F .

The next lemma from [10, Proposition 5.1.7] is fundamental.

Lemma 2.4 (Convergence Lemma). Let A be a densely defined strip-type operatorof height ω0 on a Banach space X. Let ω > ω0 and (fj)j∈J ⊆ H∞(Stω) be a netsatisfying the following conditions:

(1) supj∈J ‖fj‖H∞(Stω) <∞;

(2) f(z) := limj fj(z) exists for all z ∈ Stω;(3) supj∈J ‖fj(A)‖L(X) <∞.

Then f ∈ H∞(Stω), f(A) ∈ L(X), fj(A)→ f(A) strongly and

‖f(A)‖ ≤ lim supj∈J

‖fj(A)‖ .

2.2. Fourier multipliers on Besov spaces. Let us summarize some results aboutFourier multipliers on vector-valued Besov spaces which will be used later on. De-tails can be found in [1] and [9].

Let ψ ∈ C∞(R) be a nonnegative function with support in [ 12 , 2] such that

∞∑k=−∞

ψ(2−ks) = 1 for all s ∈ (0,∞).

For k ∈ N and s ∈ R let ϕk(s) := ψ(2−k|s|), and let ϕ0(s) := 1 −∑∞k=1 ϕk(s).

Let X be a Banach space and let p, q ∈ [1,∞] and r ∈ R be given. The (inhomo-geneous) Besov space Brp,q(R;X) consists of all X-valued tempered distributions


f ∈ S ′(R;X) such that

‖f‖Brp,q(R;X) :=∥∥∥(2kr

∥∥F−1ϕk ∗ f∥∥

Lp(R;X)

)∞k=0

∥∥∥`q<∞,

endowed with the norm ‖f‖Brp,q(R;X). Then Brp,q(R;X) is a Banach space such that

S(R;X) ⊆ Brp,q(R;X), and a different choice of ψ leads to an equivalent norm onBrp,q(R;X).

For n ∈ N and p ∈ [1,∞] the Sobolev space

Wn,p(R;X) :=f ∈ Lp(R;X) | f (k) ∈ Lp(R;X) for all 1 ≤ k ≤ n

,

is a Banach space when endowed with the norm

‖f‖n,p := ‖f‖Wn,p(X) := ‖f‖p + ‖f (n)‖p (f ∈Wn,p(R;X)).

In the case X = C we simply write Wn,p(R) = W1,p(R;C).The following lemma is equation (5.9) in [1]. The fact that the constant C does

not depend on the particular Banach space follows from a direct sum argument.

Lemma 2.5. Let θ ∈ (0, 1), p ∈ [1,∞), q ∈ [1,∞] and n ∈ N. Then there exists aconstant C > 0 such that, for any Banach space X, (Lp(X),Wn,p(X))θ,q = Bnθp,q(X)with

1

C‖f‖Bnθp,q(X) ≤ ‖f‖(Lp(X),Wn,p(X))θ,q

≤ C ‖f‖Bnθp,q(X) (f ∈ Bnθp,q(X)).

Let m ∈ L∞(R;L(X)), p, q ∈ [1,∞] and r ∈ R. We say that m is a boundedFourier multiplier on Brp,q(X) if there is a unique bounded operator Tm : Brp,q(X)→Brp,q(X) such that

Tm(f) = F−1 (m · Ff)(2.4)

for all f ∈ S(X). Each µ ∈ M(R) induces a bounded Fourier multiplier Fµ with

TFµ(f) = Lµ(f) := µ ∗ f (f ∈ S(X)).(2.5)

The main result about Fourier multipliers on Besov spaces that we use is the fol-lowing, Corollary 4.15 from [9].

Proposition 2.6. There exists a constant C ≥ 0 such that the following holds.Let X be a Banach space, p, q ∈ [1,∞] and r ∈ R. If m : R → C is such that

ϕkm ∈ B1/22,1 (R;C) for all k ∈ N0, and

M := supk∈N0

infa>0‖(ϕkm)(a·)‖

B1/22,1 (R;C)

<∞,

then m is a bounded Fourier multiplier on Brp,q(X) with ‖Tm‖L(Brp,q(X)) ≤ CM .

Corollary 2.7. There exists a constant C ≥ 0 such that for all Banach spaces X,p, q ∈ [1,∞], r ∈ R and all m ∈ C1(R;C) with

N := sups∈R|m(s)|+ (1 + |s|)|m′(s)| <∞,

m is a bounded Fourier multiplier on Brp,q(X) with ‖Tm‖L(Brp,q(X)) ≤ CN .

Proof. This follows as in [9, Corollary 4.11]. See also [9, Remark 4.16].


3. Transference principles

3.1. Unbounded groups. We first establish an interpolation version of the trans-ference principle for unbounded groups from [12]. Note that, for each µ ∈ M(R) andp ∈ [1,∞], the convolution operator Lµ from (2.5) extends to a bounded operatoron Lp(X), by Young’s inequality. For ω ≥ 0 and µ ∈ Mω(R) let µω ∈ M(R) begiven by µω(ds) := cosh(ωs)µ(ds).

Proposition 3.1. Let 0 ≤ ω0 < ω, θ ∈ (0, 1), p ∈ [1,∞) and q ∈ [1,∞]. Thenthere exists a constant C ≥ 0 such that the following holds. If −iA generates aC0-group (U(s))s∈R on a Banach space X such that ‖U(s)‖L(X) ≤M cosh(ω0s) for

all s ∈ R and some M ≥ 1, then∥∥∥∥∫RU(s)xµ(ds)

∥∥∥∥θ,q

≤ CM2 ‖Lµω‖L(Bθp,q(X)) ‖x‖θ,q

for all µ ∈ Mω(R) and x ∈ DA(θ, q).

Proof. Let µ ∈ Mω(R) be given and let Uµ be as in (2.2). By the proof of Theorem3.2 in [12], we can factorize Uµ as Uµ = P Lµω ι, where

• ι : X → Lp(X) is given by

ιx(s) := ψ(−s)U(−s)x (x ∈ X, s ∈ R),

with

ψ(s) :=1

cosh(αs)(s ∈ R)

for α > ω fixed.• P : Lp(X)→ X is given by

Pf :=

∫Rϕ(s)U(s)f(s) ds (f ∈ Lp(X)),

with

ϕ(s) :=

√8ω

π

cosh(ωs)

cosh(2ωs)(s ∈ R).

Then, using Holder’s inequality,

‖ι‖L(X,Lp(X)) ≤M ‖ψ cosh(ω0·)‖p ,(3.1)

‖P‖L(Lp(X),X) ≤M ‖ϕ cosh(ω0·)‖p′ .(3.2)

We claim that ι : D(A)→W1,p(X) and P : W1,p(X)→ D(A) are well-defined andbounded. To prove this claim, first let x ∈ D(A). Then ιx ∈ C1(R) with

(ιx)′(s) = −ψ′(−s)U(−s)x+ iψ(−s)U(−s)Ax

= −α tanh(αs)

cosh(αs)U(−s)x+ i

1

cosh(αs)U(−s)Ax

for all s ∈ R. Hence (ιx)′ ∈ Lp(R) with

‖(ιx)′‖p ≤ αM ‖tanh‖L∞(R)

∥∥∥∥cosh(ω0·)cosh(α·)

∥∥∥∥p

‖x‖X +M


∥∥∥∥p

‖Ax‖X .


Combining this with (3.1) implies that ιx ∈W1,p(R) with

‖ιx‖1,p ≤M(α ‖tanh‖L∞(R) + 1)


∥∥∥∥p

‖x‖D(A) .(3.3)

This shows that ι : D(A) → W1,p(X) is bounded. To prove the claim for P , fixf ∈ S(X) and note that

1

h(U(h)− I)Pf =

∫RU(s)

ϕ(s− h)f(s− h)− ϕ(s)f(s)

hds

for h > 0. The latter expression converges to −∫R U(s)(ϕf)′(s) ds ∈ X as h → 0,

by the dominated convergence theorem. Hence Pf ∈ D(A) with

APf = limh→0

1

h(U(h)− I)Pf = −

∫RU(s)(ϕ′(s)f(s) + ϕ(s)f ′(s)) ds.

Another application of Holder’s inequality yields

‖APf‖X ≤M ‖ϕ′ cosh(ω0·)‖p′ ‖f‖p +M ‖ϕ cosh(ω0·)‖p′ ‖f

′‖p .

Combining this with (3.2) implies

‖Pf‖D(A) ≤M(‖ϕ cosh(ω0·)‖p′ + ‖ϕ′ cosh(ω0·)‖p′

)‖f‖1,p .(3.4)

As S(X) is dense in W1,p(X), P : W1,p(X)→ D(A) is bounded.Since Lµω ∈ L(W1,p(X)), we can factorize Uµ ∈ L(D(A)) as Uµ = P Lµω ι

via bounded maps through W1,p(X). Applying the real interpolation method withparameters θ and q to the two factorizations of Uµ, through Lp(X) respectivelyW1,p(X), yields the commutative diagram of bounded maps

(Lp(X),W1,p(X))θ,qLµω−−−−→ (Lp(X),W1,p(X))θ,qxι yP

DA(θ, q)Uµ−−−−→ DA(θ, q)

Finally, estimate the norms of ι and P in this diagram by applying (1.5) to (3.1)and (3.3) respectively (3.2) and (3.4). This yields

‖Uµ‖L(DA(θ,q)) ≤ C′M2 ‖Lµ‖L((Lp(X),W1,p(X))θ,q)

(3.5)

for a constant C ′ ≥ 0 independent of µ. Now Lemma 2.5 concludes the proof.

3.2. Bounded groups. In this section we establish a version of the classical trans-ference principle from [5] on interpolation spaces, already stated in the Introductionas Proposition 1.1. In the proof we use the convention 1/∞ := 0.

Proposition 3.2. Let θ ∈ (0, 1), p ∈ [1,∞) and q ∈ [1,∞]. Then there exists aconstant C ≥ 0 such that the following holds. If −iA generate a C0-group (U(s))s∈Ron a Banach space X with M := sups∈R ‖U(s)‖ <∞, then∥∥∥∥∫

RU(s)xµ(ds)

∥∥∥∥θ,q

≤ CM2 ‖Lµ‖L(Bθp,q(X)) ‖x‖θ,q(3.6)

for all µ ∈ M(R) and x ∈ DA(θ, q).


Proof. First note that it suffices to establish (3.6) for measures with compact sup-port. Indeed, approximating by measures with compact support then extends (3.6)to all µ ∈ M(R). So fix N > 0 and let µ ∈ M(R) be such that supp(µ) ⊆ [−N,N ].We will factorize Uµ using the abstract transference principle from [13, Section 2].To this end, let ρ ∈ C∞(R) be defined by

ρ(s) :=

c1 exp

(1

s2−1

)|s| < 1

0 |s| ≥ 1,

where c1 ≥ 0 is such that∫R ρ(s) ds = 1. Fix α, β > 0 and define σ(s) := 1

α ρ(sα

)for s ∈ R, and

ψ := σ ∗ 1[−(N+3α+β),N+3α+β] and ϕ :=1

2(α+ β)σ ∗ 1[−(α+β),α+β].

Then ψ,ϕ ∈ C∞(R) are such that supp(ϕ) ⊆ [−(2α+ β), 2α+ β],

ψ ≡ 1 on [−(2α+N + β), 2α+N + β] and

∫ 2α+β

−(2α+β)

ϕ(s) ds = 1.

Hence ψ ∗ ϕ ≡ 1 on [−N,N ]. Let ι : X → Lp(X) be given by

ιx(s) := ψ(−s)U(−s)x (x ∈ X, s ∈ R),

and P : Lp(X)→ X by

Pf :=

∫Rϕ(s)U(s)f(s) ds (f ∈ Lp(X)).

Proposition 2.3 in [13] yields the factorization Uµ = P Lµ ι, where we use that(ψ ∗ ϕ)µ = µ. By Holder’s inequality,

‖ι‖L(X,Lp(X)) ≤M ‖ψ‖p and ‖P‖L(Lp(X),X) ≤M ‖ϕ‖p′(3.7)

Moreover, ι : D(A)→W1,p(X) and P : W1,p(X)→ D(A) are bounded with

‖ι‖L(D(A),W1,p(X)) ≤M ‖ψ‖1,p and ‖P‖L(W1,p(X),D(A)) ≤M ‖ϕ‖1,p′ .(3.8)

This follows by arguments almost identical to those in the proof of Proposition 3.1.Applying the real interpolation method with parameters θ and q to the two factor-izations of Uµ, through Lp(X) and W1,p(X), produces the commutative diagramof bounded maps

(Lp(X),W1,p(X))θ,qLµ−−−−→ (Lp(X),W1,p(X))θ,qxι yP

DA(θ, q)Uµ−−−−→ DA(θ, q)

Use (1.5) on (3.7) and (3.8) to estimate the norms of ι and P in this factorizationas ‖ι‖ ≤M ‖ψ‖1,p and ‖P‖ ≤M ‖ϕ‖1,p′ . This yields

‖Uµ‖L(DA(θ,q)) ≤M2 ‖ψ‖1,p ‖ϕ‖1,p′ ‖Lµ‖L((Lp(X),W1,p(X))θ,q)

.(3.9)

To determine ‖ψ‖1,p and ‖ϕ‖1,p′ , note that

‖ψ‖p ≤ ‖σ‖1∥∥1[−(N+3α+β),N+3α+β]

∥∥p

= (2(N + 3α+ β))1/p,

‖ϕ‖p′ ≤1

2(α+ β)‖σ‖1

∥∥1[−(α+β),α+β]

∥∥p′

= (2(α+ β))−1/p,


by Young’s inequality. Since σ is an even function that is decreasing on [0, α] andsupported on [−α, α], its derivative satisfies

‖σ′‖1 = −2

∫ α

0

σ′(s) ds = 2(σ(0)− σ(α)) =2ρ(0)

α.

Let c2 := 2ρ(0). Another application of Young’s inequality yields

‖ψ′‖p ≤ ‖σ′‖1∥∥1[−(N+3α+β),N+3α+β]

∥∥p

=c2α

(2(N + 3α+ β))1/p,

‖ϕ′‖p ≤1

2(α+ β)‖σ′‖1

∥∥1[−(α+β),α+β]

∥∥p

=c2α

(2(α+ β))−1/p.

Hence (3.9) becomes

‖Uµ‖L(DA(θ,q)) ≤M2(

1 +c2α

)2(N + 3α+ β

α+ β

)1/p

‖Lµ‖L((Lp(X),W1,p(X))θ,q).

Taking the infimum over α and β yields

‖Uµ‖L(DA(θ,q)) ≤M2 ‖Lµ‖L((Lp(X),W1,p(X))θ,q)

.(3.10)

Lemma 2.5 now establishes (3.6) and concludes the proof.

Remark 3.3. Note that the constant C in Proposition 3.2 comes only from theequivalence of the norms on (Lp(X),W1,p(X))θ,q and Bθp,q(X), whereas in Propo-sition 3.1 a constant is present which is inherent to the transference method.

Remark 3.4. Let p ∈ [1,∞) and let (U(s))s∈R ⊆ L(Lp(C)) be the shift groupgiven by U(s)f(t) := f(t + s) for f ∈ Lp(C), s ∈ R and almost all t ∈ R. Then(U(s)))s∈R is generated by −iA, where Af := if ′ for f ∈ D(A) = W1,p(C). HenceDA(θ, q) = (Lp(C),W1,p(C))θ,q for θ ∈ (0, 1) and q ∈ [1,∞]. Moreover, for µ ∈M(R) and f ∈ Lp(C), ∫

RU(s)f dµ(s) = µ ∗ f = Lµ(f).

Hence, with Uµ as in (2.2),

‖Uµ‖L(DA(θ,q)) = ‖Lµ‖L((Lp(C),W1,p(C))θ,q).

This shows that (3.10) is sharp in general, up to possibly a change of constant. ByLemma 2.5, the same holds for (3.6).

Corollary 2.7 yields the following result, Corollary 1.2 from the Introduction.

Corollary 3.5. Let θ ∈ (0, 1) and q ∈ [1,∞]. Then there exists a constant C ≥ 0such that the following holds. Let −iA generate a C0-group (U(s))s∈R on a Banachspace X with M := sups∈R ‖U(s)‖ <∞, and let µ ∈ M(R) be such that Fµ ∈ C1(R)with sups∈R(1 + |s|) |(Fµ)′(s)| <∞. Then∥∥∥∥∫

RU(s)xµ(ds)

∥∥∥∥θ,q

≤ CM2 sups∈R|Fµ(s)|+ (1 + |s|) |(Fµ)′(s)| ‖x‖θ,q

for all x ∈ DA(θ, q).


Remark 3.6. To obtain Corollary 3.5 we used Corollary 2.7, but there are otherways to verify the conditions of Proposition 2.6, for instance Hormander type as-sumptions, cf. [9, pp. 47-49]. More generally, one can define a norm on the space ofall bounded Fourier multipliers m on Brp,q(X) by ‖m‖M(Brp,q(X)) := ‖Tm‖L(Brp,q(X)),

with Tm as in (2.4). Proposition 3.2 yields ‖Uµ‖L(DA(θ,q)) ≤ C ‖Fµ‖M(Brp,q(X)),

which cannot be improved in general, cf. Remark 3.4.

Remark 3.7. If X is a UMD space then (1.1) and the vector-valued Mikhlinmultiplier theorem [10, Theorem E.6.2 b] yield an estimate∥∥∥∥∫

RU(s)xµ(ds)

∥∥∥∥X

≤ CM2 ‖x‖X sups∈R|Fµ(s)|+ |s(Fµ)′(s)|

for all x ∈ X. Corollary 3.5 then follows from (1.5), and moreover singularities of(Fµ)′ at zero are allowed. However, in our setting of general Banach spaces, theinhomogeneity of the Besov space Brp,q(X) implies that a condition at zero on themultiplier is needed to deal with the term ϕ0m in Proposition 2.6.

Remark 3.8. Letting f := Fµ, Corollary 3.5 yields an estimate

‖f(Aθ,q)‖ ≤ C sups∈R|f(s)|+ (1 + |s|) |f ′(s)|.(3.11)

This is a functional calculus statement for Aθ,q involving functions on the real line.One may now ask to which functions f on the real line the definition of f(Aθ,q) canbe extended in a sensible manner such that (3.11) holds. We can take the closure ofthe Fourier transforms of measures in the space consisting of all functions f ∈ C1(R)for which sups∈R |f(s)| + (1 + |s|) |f ′(s)| is finite, or approximate by holomorphicfunctions as in [18, Lemma 4.15], using Theorem 4.1. This will yield a definitionof f(Aθ,q) for a class of functions on the real line and a bound as in (3.11), butthe question then remains how this definition relates to other known extensionsof functional calculi. In the present article we restrict ourselves to results aboutholomorphic functional calculi.

4. Functional calculus results

We now use the theory established in the previous sections to prove our mainfunctional calculus result, Theorem 1.3. Recall the definition of the analytic Mikhlinalgebra H∞1 (Stω) from (1.2).

Theorem 4.1. Let −iA be the generator of a C0-group (U(s))s∈R on a Banachspace X and let θ ∈ (0, 1), q ∈ [1,∞] and ω > θ(U) be given. Then there exists aconstant C ≥ 0 such that f(Aθ,q) ∈ L(DA(θ, q)) with

‖f(Aθ,q)‖L(DA(θ,q)) ≤ C ‖f‖H∞1 (Stω)

for all f ∈ H∞1 (Stω). If (U(s))s∈R is uniformly bounded then C can be chosenindependent of ω > 0.

Proof. First consider f ∈ H∞1 (Stω) ∩ E(Stω) and fix α ∈ (θ(U), ω) and p ∈ [1,∞).By Lemma 2.1 there exists µ ∈ Mα(R) such that f = Fµ. By Lemmas 2.1 and 2.2and Proposition 3.1,

‖f(Aθ,q)‖ = ‖(Uµ)θ,q‖ ≤ C1 ‖Lµα‖L(Bθp,q(X)) = C1 ‖TFµα‖L(Bθp,q(X))(4.1)


for some constant C1 ≥ 0, where TFµα is as in (2.4). Since

Fµα(s) =f(s+ iα) + f(s− iα)

2(s ∈ R),

Corollary 2.7 yields a constant C2 ≥ 0 such that

‖f(Aθ,q)‖ ≤ C2 sups∈R|Fµα(s)|+ (1 + |s|)|(Fµα)′(s)| ≤ C2 ‖f‖H∞

1 (Stω) .(4.2)

For general f ∈ H∞1 (Stω) first assume that q < ∞. By part b) of Lemma 2.2,D(Aθ,q) is dense in DA(θ, q). Let τk(z) := −k2(ik− z)−2 for k ∈ N with k > ω andz ∈ Stω. Then τk, fτk ∈ H∞1 (Stω) ∩ E(Stω),

supk‖fτk‖H∞

1 (Stω) ≤ ‖f‖H∞1 (Stω) sup

k‖τk‖H∞

1 (Stω) <∞

and fτk(z)→ f(z) as k →∞, for all z ∈ Stω. Now (4.2) yields

‖fτk(Aθ,q)‖ ≤ C2 ‖fτk‖H∞1 (Stω) ≤ C ‖f‖H∞

1 (Stω)

for some C ≥ 0. Hence the Convergence Lemma 2.4 implies f(A) ∈ L(X) and

‖f(Aθ,q)‖ ≤ C ‖f‖H∞1 (Stω) .(4.3)

Finally, for q =∞ the Reiteration Theorem [4, Theorem 3.5.3] yields

DA(θ,∞) = (DA(θ1, 1),DA(θ2, 1))θ3,∞

with equivalence of norms, where θ1, θ2, θ3 ∈ (0, 1) are such that θ1 6= θ2 andθ1(1 − θ3) + θ2θ3 = θ. Combining (4.3) and (1.5) concludes the proof of the firststatement.

In the case where (U(s))s∈R is uniformly bounded, use Proposition 3.2 insteadof 3.1 in (4.1) to obtain

‖f(Aθ,q)‖ ≤ C1 ‖TFµ‖L(Bθp,q(X))

for all f ∈ H∞1 (Stω)∩E(Stω) and some constant C1 ≥ 0 independent of ω. The restof the proof is the same as before.

Remark 4.2. Compare Theorem 4.1 with Theorem 3.6 in [12]. There an estimate

‖f(A)‖L(X) ≤ C supz∈Stω

|f(z)|+ |zf ′(z)|(4.4)

is obtained when the underlying space X is a UMD space, and the constant Cis independent of ω when the group in question is uniformly bounded. Theo-rem 4.1 follows from (4.4) by interpolation, and this seems to yield a strongerresult since the term supz∈Stω |f

′(z)| does not appear in (4.4). In fact, the normssupz∈Stω |f(z)|+ |zf ′(z)| and ‖f‖H∞

1 (Stω) are equivalent, since 0 ∈ Stω for all ω > 0.

So for generators of unbounded groups (4.4) does not yield an essentially betterestimate than Theorem 4.1. This is different for generators of uniformly boundedgroups, since the norm equivalence of supz∈Stω |f(z)|+ |zf ′(z)| and ‖f‖H∞

1 (Stω) fails

as ω ↓ 0. Hence for generators of uniformly bounded groups (4.4) yields a strictlystronger result on DA(θ, q) than Theorem 4.1.

Remark 4.3. Let λ ∈ C with Re(λ) > ω. By [10, Corollary 6.6.3], D((λ− iA)α) ⊆DA(α,∞) for each α ∈ (0, 1). Hence Theorem 4.1 yields f(A)(λ − iA)−α ∈ L(X)for all ω > θ(U), f ∈ H∞1 (Stω) and α > 0. However, this already follows from [3,Proposition 8.2.3] in a similar manner as in [15, Remark 5.2]. Moreover, usingarguments as in [15, Remark 3.9], [3, Proposition 8.2.3] already implies that f(A) :


DA(θ, q)→ DA(θ′, q′) is bounded for all θ′ < θ and q, q′ ∈ [1,∞]. The improvementthat Theorem 4.1 provides lies in going from θ′ < θ to θ′ = θ.

Remark 4.4. As already noted in Remark 3.6, we could have used Fourier mul-tiplier results on Besov spaces other than Corollary 2.7. These lead to statementsabout the boundedness of functional calculi for other function algebras.

For ϕ ∈ (0, π) define

Sϕ := z ∈ C | |arg(z)| < ϕ ,(4.5)

and for ψ ∈ (0, π/2) and ω > 0,

Σψ := Sψ ∪ −Sψ, Vψ,ω := Stω ∪ Σψ.

Lemma 4.5. Let ω > ω′ > 0 and ψ ∈ (0, π/2). Then H∞(Vω,ψ) is continuouslyembedded in H∞1 (Stω′).

Proof. This follows in a straightforward manner from Lemma 4.5 in [12].

Corollary 4.6. Let −iA be the generator of a C0-group (U(s))s∈R on a Banachspace X and let θ ∈ (0, 1) and q ∈ [1,∞]. Then Aθ,q has a bounded H∞(Vω,ψ)-calculus for all ω > θ(U) and ψ ∈ (0, π/2).

So far we have considered functional calculus on interpolation spaces for thecouple (X,D(A)). The next corollary extends our results to other interpolationcouples.

Corollary 4.7. Let −iA be the generator of a C0-group (U(s))s∈R on a Banachspace X and let θ ∈ (0, 1), q ∈ [1,∞] and m,n ∈ N0 with m 6= n. Then the partof A in (D(Am),D(An))θ,q has a bounded H∞1 (Stω)-calculus for all ω > θ(U). If(U(s))s∈R is uniformly bounded then the constant bounding the calculus is indepen-dent of ω > 0.

Proof. First note that since

(D(Am),D(An))θ,q = (D(An),D(Am))1−θ,q

by [4, Theorem 3.4.1], we may assume that m < n. Using the similarity transformR(λ,A)m : X → D(Am), it suffices to let m = 0. Suppose that nθ /∈ N. By Lemma3.1.3 and Proposition 3.1.8 in [19],

(X,D(An))θ,q = (D(Ak),D(Ak+1))θ′,q

for some k ∈ N0 and θ′ ∈ (0, 1). Another similarity transform shows that we canlet k = 0. Now Theorem 4.1 yields the statement.

If k := nθ ∈ N, the Reiteration Theorem [4, Theorem 3.5.3] yields

(X,D(An))θ,q =((D(Ak−1),D(Ak))1/2,q, (D(Ak),D(Ak+1))1/2,q

)1/2,q

.

By what we have already shown and (1.5), this concludes the proof.


5. Additional results

We now deduce several applications of Theorem 4.1. Corollary 4.7 can be appliedin this section to yield results for other interpolation couples.

We first state a proposition about the convergence of certain principal valueintegrals, an interpolation version of [12, Theorem 4.4] on general Banach spaces.If g ∈ L1[−1, 1] is even then by PV − g(s)/s we mean the distribution defined by

〈PV − g(s)/s, ϕ〉 := limε0

∫ε≤|s|≤1

g(s)ϕ(s)ds

s

for ϕ ∈ C∞(R) compactly supported. By BV[−1, 1] we denote the functions ofbounded variation on [−1, 1].

Proposition 5.1. Let −iA be the generator of a C0-group (U(s))s∈R on a Banachspace X. Let g ∈ BV[−1, 1] be even and set f := F(PV− g(s)/s). Then f(Aθ,q) ∈L(DA(θ, q)) and

f(A)x = limε0

∫ε≤|s|≤1

g(s)U(s)xds

s(5.1)

for all θ ∈ (0, 1), q ∈ [1,∞) and x ∈ DA(θ, q).

Proof. By [12, Lemma 4.3], f ∈ H∞1 (Stω) for all ω > 0. Theorem 4.1 now yieldsthe first statement. For (5.1) we may let q < ∞, since DA(θ,∞) ⊆ DA(θ′, 1) forθ′ < θ [19, Proposition 1.1.4]. Now use the Convergence Lemma as in the proofof [12, Theorem 4.4].

Remark 5.2. Convergence in (5.1) takes place in DA(θ, q) for q <∞. For q =∞the limit and the integral converge in X and in DA(θ′, q′) for θ′ < θ and q′ ∈ [1,∞).Compare with Remark 2.3.

5.1. Results for sectorial operators and cosine functions. An operator A ona Banach space X is sectorial of angle ϕ ∈ (0, π) if σ(A) ⊆ Sϕ, where Sϕ is as in(4.5), and if sup ‖zR(z,A)‖ | ψ ∈ C \ Sψ < ∞ for all ψ ∈ (ϕ, π). A functionalcalculus for sectorial operators can be constructed by a method similar to the oneused for strip-type operators. For details see [10, Chapter 2].

If A is an injective sectorial operator of angle ϕ ∈ (0, π) then log(A) is defined,as is f(A) for all f ∈ H∞(Sψ) and ψ ∈ (ϕ, π). A sectorial operator A has boundedimaginary powers if A is injective and if −i log(A) is the generator of a C0-group(U(s))s∈R on X. Then U(s) = A−is for all s ∈ R, and we write A ∈ BIP(X).Moreover, A is sectorial of angle θA := θ(U), by [10, Corollary 4.3.4].

For ψ ∈ (0, π) define H∞log(Sψ) to be the unital Banach algebra of all f ∈ H∞(Sψ)for which

‖f‖H∞log(Sψ) := sup

z∈Sψ

|f(z)|+ (1 + |log(z)|)|zf ′(z)| <∞,

endowed with the norm ‖·‖H∞log(Sψ).

Proposition 5.3. Let X be a Banach space and A ∈ BIP(X) such that θA < π.Let θ ∈ (0, 1) and q ∈ [1,∞]. Set Y := (X,D(log(A)))θ,q. Then AY has a boundedH∞log(Sψ)-calculus on Y for all ψ ∈ (θA, π). If sups∈R

∥∥Ais∥∥ <∞ then the constant

bounding the calculus is independent of ψ > 0.


Proof. Let ψ ∈ (θA, π) be given and note that f 7→ f log is an isometric algebraisomorphism H∞1 (Stψ) → H∞log(Sψ). By Lemma 2.2 as well as Theorem 4.2.4 and

Proposition 6.1.2 from [10],

f(log(A)Y ) = f(log(A))Y = (f log)(A)Y = (f log)(AY )

for all f ∈ H∞1 (Stψ). Now Theorem 4.1 concludes the proof.

Remark 5.4. Let A be an injective sectoral operator of angle ϕ ∈ (0, π), andlet α > 0, θ ∈ (0, 1) and q ∈ [1,∞]. By [10, Corollary 6.6.3], a special case ofwhich was proved by Dore [7, Theorem 3.2], the part of A in (X,D(Aα)∩R(Aα))θ,qhas a bounded H∞(Sψ)-calculus for all ψ ∈ (ϕ, π). Here R(A) is the range of A.By [10, Corollary 6.6.3] and because log(A)Aαθ(1 +A)−2αθ ∈ L(X),

(X,D(Aα) ∩ R(Aα))θ,q ⊆ (X,D(Aα))θ,q ⊆ D(Aαθ) ⊆ D(log(A)),

and in general D(log(A)) is strictly included in (X,D(log(A)))θ′,q′ for all θ′ ∈ (0, 1)and q′ ∈ [1,∞]. Hence the result of Dore does not imply Proposition 5.3.

A cosine function Cos : R→ L(X) on a Banach space X is a strongly continuousmapping such that Cos(0) = I and

Cos(t+ s) + Cos(t− s) = 2Cos(t)Cos(s)

for all s, t ∈ R. Then

θ(Cos) := infω ≥ 0

∣∣∣∃M ≥ 0 : ‖Cos(t)‖ ≤Meω|t| for all t ∈ R<∞.

The generator of a cosine function is the unique operator −A on X that satisfies

λR(λ2,−A) =

∫ ∞0

e−λtCos(t) dt

for λ > θ(Cos). Then A is an operator of parabola-type ω = θ(Cos). This meansthat σ(A) ⊆ Πω, where Πω :=

z2 | z ∈ Stω

, and that for all ω′ > ω there exists

Mω′ ≥ 0 such that

‖R(λ,A)‖ ≤ Mω′√|λ|(|Im(√λ)| − ω′

)for all λ /∈ Πω′ . For such operators there is a natural functional calculus, as before,and a version of Lemma 2.2 holds. For details see [14]. For ω > 0 let

H∞1 (Πω) :=

f ∈ H∞(Πω)

∣∣∣∣‖f‖H∞1 (Πω) := sup

z∈Πω|f(z)|+ (1 + |z|) |f ′(z)| <∞

,

a Banach algebra when endowed with the norm ‖·‖H∞1 (Πω).

Proposition 5.5. Let −A be the generator of a cosine function Cos on a Banachspace X and let θ ∈ (0, 1), q ∈ [1,∞]. Then the part Aθ,q of A in DA(θ, q) hasa bounded H∞1 (Πω)-calculus for all ω > θ(Cos). If sups∈R ‖Cos(s)‖ < ∞ then theconstant bounding the calculus is independent of ω > 0.

Proof. We mainly follow [12, Theorem 5.3], providing extra details where necessary.There is a unique subspace V ⊆ X, the Kisynski space, such that the operator −iA,

A := i

[0 IV−A 0

],


with domain D(A) := D(A)×V , generates a C0-group (U(s))s∈R on X ×V . More-over, θ(Cos) = θ(U) [11, Theorem 6.2].

Let ω > θ(Cos). Then f ∈ H∞(Πω) is an element of H∞1 (Πω) if and onlyif [z 7→ g(z) := f(z2)] ∈ H∞1 (Stω), with ‖g‖H∞

1 (Stω) ≤ 4 ‖f‖H∞1 (Πω). Moreover,

f(A)⊕ f(AV ) = g(A) and

f(Aθ,q)⊕ f(AV ) = (f(A)⊕ f(AV ))Y = g(A)Y = g(AY )

for all f ∈ H∞1 (Πω), where Y := DA(θ, q) = DA(θ, q) × V . Hence Theorem 4.1concludes the proof.

Acknowledgements. We would like to thank our colleagues for interesting discus-sions and helpful ideas.

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Delft Institute of Applied Mathematics, Mekelweg 4, 2628CD Delft, The Nether-

landsE-mail address: [email protected]

Delft Institute of Applied Mathematics, Mekelweg 4, 2628CD Delft, The Nether-lands

E-mail address, corresponding author: [email protected]

Functional calculus for $C_{0}$-groups on real interpolation spaces

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